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The purpose of this workshop is to link mathematicians working in operator theory and noncommutative calculus with mathematical physicists in the area of the spectral action principle.

The Spectral Action is a new and high profile theory that unites quantum and classical action principles in a mathematically elegant construction. Professor Walter van Suijlekom (the Institute for Mathematics, Astrophysics and Particle Physics of the Radboud University Nijmegen) will give a mini-lecture series on the spectral actions during the mornings of the workshop. Talks on the spectral action will be given by other international experts, including Professor Giovanni Landi and Professor Bruno Iochum. Specific mathematical topics are trace class aysmptotic expansions, noncommutative Taylor series expansions, Dixmier trace and higher order residues.

This is to announce a series of lectures delivered by Ali Chamseddine (American University of Beirut, Lebanon), who is visiting Radboud University Nijmegen this Spring as a Radboud Excellence Professor. All talks will take place in the Huygensgebouw (Science Faculty) of Radboud University Nijmegen.

The first lecture (on March 26, 2019, 15:30-16:30, HG00.622) will be intended for a broad audience (with some background/interest in physics), and will be on the noncommutative geometry underlying the Standard Model of Particle Physics. Afterwards there will be a reception.

In the second and third lecture (on April 2 and 9, 2019, 15:30-16:30 in HG00.622 and HG00.108, resp.) the formalism will be further laid out; these lectures will be more specialized. Below you find more detailed titles and abstracts.

Abstract (Lecture 1)
The fundamental particles and their interactions are well represented by the the Standard Model of
Particle Physics and Einstein gravity. It is not known or understood the reasons why nature made this
choice for the particles and their representations. In this lecture, following down to top approach, I will
show that noncommutative geometry, as defined by Alain Connes, is the underlying geometry of space-time
and provides the answer to “Why the Standard Model”.

Abstract (Lecture 2)
Addressing the mathematical question of characterizing four-dimensional spin manifolds leads to a
relation similar to Heisenberg noncommutativity of coordinates and momenta in Quantum Mechanics.
I show that this relation implies the quantization of the volume of space-time and leads uniquely to
identify the underlying geometry, which turns out to be noncommutative. The resulting dynamical theory
unifies all interactions including gravity with the Standard Model the effective symmetry at low energies.

Abstract (Lecture 3)
Over the years many attempts were made to isolate the scale factor in General Relativity (known as
the dilaton) but these, however, proved not to be useful. A novel idea is presented where the scale
factor is isolated in a covariant way where it will be traded for a scalar field representing the time coordinate.
I will show that this innocent looking modification has drastic consequences on cosmology. In particular, geometric
invariants of the scalar field now mimic dark matter without the need for new interactions. These can also provide
mechanisms to avoid singularities for Friedman and Kasner cosmologies, as well as black hole singularities of
Schwarzschild like metrics.

Subfactors and Quantum Groups: a combinatorial approach to some problems in Operator Algebras

Masterclass, University of Copenhagen, 29 April – 3 May 2019

The celebrated theory of subfactors studied by V. Jones in the early 1980’s finds a connection with discrete quantum groups (duals of compact quantum groups in the sense of Woronowicz) through the concept of rigid C*-tensor category thanks to works of S. Popa in the 1990’s. The aim of this 5-days masterclass is to provide a comprehensive introduction to the subfactor theory as well as the combinatorics behind in order to understand this connection and highlight some recent applications.

The masterclass will consist of three lecture series by Michael Brannan, Amaury Freslon and Stefaan Vaes, accompained by several problem sessions. Besides, there will be three contributed talks which will complement the topics treated in the mini-courses.

The Mathematics department of Radboud University (Nijmegen, The Netherlands) has a vacancy for a tenure track assistant/associate professorship in Geometry (broadly interpreted).

Initially this concerns an employment for six years. There will be a final evaluation after at most five and a half years. If the final review is positive the position will be converted to a permanent one. For candidates who are already further in their career an earlier evaluation is possible.

The department aims at increasing the number of female academic mathematicians employed by the institute. Female candidates are therefore particularly encouraged to apply.

Today my new preprint with Ali Chamseddine and Alain Connes appeared on “entropy and the spectral action”. In this paper we compute the information theoretic von Neumann entropy of the state associated to the fermionic second quantization of a spectral triple and show that it is given by a spectral action of the spectral triple for a specific universal function. We find a surprising relation between this function and the Riemann zeta function.

In the paper we pass from the one-particle level of spectral triples \((\mathcal A, \mathcal H,D)\) to a fermionic second-quantized level according to the following dictionary:

There is then a unique KMS\({}_\beta\)-state associated to the \(C^*\)-dynamical system \( ( {\rm Cliff}_{\mathbb C}(\mathcal H_{\mathbb R} ), \sigma_t )\). We show that the von Neumann information theoretic entropy of this state is equal to the spectral action \({\rm Trace}( \mathcal E(e^{-\beta D}))\) where \(\mathcal E(x) = \log (x+1) – \frac {x \log x}{x+1}\) (note that the latter expression is the entropy of the partition of the unit interval in two intervals with ratio of size \(x\)). The function \( \mathcal E(e^{-x})\) looks like:

In the second part of the paper we analyze the structure of this function and show that it is a Laplace transform. Hence we can exploit heat kernel techniques that might be available for the heat kernel \({\rm Trace} (e^{-t D^2}) \). We then establish that the coefficients \(\gamma(a)\) of \(t^a\) in an asymptotic heat expansion for the entropy/spectral action are given by \(\gamma(a) = \frac{1-2^{-2a}}{a} \pi^{-a} \xi (2a) \) in terms of the Riemann xi function. We have listed a few of these values below:

What this table of values also shows is that the functional equation gives a duality between the coefficients of the high energy expansion in even dimension with the coefficients of the low energy expansion in the odd dimensional case.